The amount of data used in optical tomography image reconstruction has increased by several orders of magnitude in recent years. This is primarily due to the use of large detector arrays, e.g., on the order of 103 elements or higher. When coupled with a large number of sources, e.g., on the order of 102 sources (such large number being facilitated, for example, by the use of non-contact measurements) large data sets easily in the range of 105 source-detector pairs are generated. These large data sets reduce the ill-posed nature of the inversion, but also present an inherently large computational burden for reconstruction of tomographic images. Using traditional real-space weight matrix and Algebraic Reconstruction Techniques (ART) for the inversion yields impractically long computational times, in some instances longer that 24 hours. Similarly, using matrix-related inversion methods such as Singular Value Decomposition is not viable due to the amount of memory required. Thus, there is a need for a different approach that can handle large data sets and still maintain reasonably low computational times.
A powerful formalism for significantly reducing the number of measurements while maintaining the same amount of useful information is to work in Fourier Space. Diffuse light in the continuous wave (CW) regime is known to present only low spatial frequency contributions. By using all real-space data while selecting only a few low-frequency components in Fourier space, it is possible to benefit from the same amount of useful information while retaining a lower number of measurements.
Certain limited Fourier space techniques have been used to solve inverse problems in the past, for example, backprojection techniques and direct inversion techniques.
Backprojection suffers from being non-quantitative, low in resolution and incapable of good depth-discrimination [Matson, C. L., N. Clark, et al. (1997). “Three-dimensional tumor localization in thick tissue with the use of diffuse photon-density waves.” Applied Optics 36: 214-220; Matson, C. L. (2002). “Diffraction Tomography for Turbid Media.” Advances in Imaging and Electron Physics 124: 253-342; Li, X. D., T. Durduran, et al. (1997). “Diffraction tomography for biochemical imaging with diffuse-photon density waves.” Optics Letters 22: 573-575; Li, X., D. N. Pattanayak, et al. (2000). “Near-field diffraction tomography with diffuse photon density waves.” Phys Rev E 61(4 Pt B): 4295-309].
Complete Fourier approaches, also termed Direct Inversion, present severe reconstruction artifacts and generally are not applicable to datasets with fewer than O(103) source positions [(Schotland, J. C. and V. A. Markel (2001). “Inverse scattering with diffusing waves.” J Opt Soc Am A Opt Image Sci Vis 18(11): 2767-77; Markel, V. A. and J. C. Schotland (2001). “Inverse problem in optical diffusion tomography. I. Fourier-Laplace inversion formulas.” J Opt Soc Am A Opt Image Sci Vis 18(6): 1336-47; Markel, V. A. and J. C. Schotland (2004). “Symmetries, inversion formulas, and image reconstruction for optical tomography.” Phys Rev E Stat Nonlin Soft Matter Phys 70(5 Pt 2): 056616; Markel, V. A. and J. C. Schotland (2001). “Inverse scattering for the diffusion equation with general boundary conditions.” Phys Rev E 64(3 Pt 2): 035601].